Wave Packet Calculation of Cross Sections, Product State Distributions, and BranchingRatios for the O(1D) + HCl Reaction

Valentina Piermarini † and Gabriel G. Balint-KurtiSchool of Chemistry, UniVersity of Bristol, Bristol, United Kingdom

Stephen K. GrayChemistry DiVision, Argonne National Laboratory, Argonne, Illinois 60439

Fahrettin Go1gtasPhysics Department, Firat UniVersity, 23119 Elazig, Turkey

Antonio Lagana* and Marıa Luz Herna ndez‡

Dipartimento di Chimica, UniVersita di Perugia, 06123 Perugia, Italy

ReceiVed: NoVember 17, 2000; In Final Form: March 30, 2001

Time-dependent quantum mechanical calculations have been carried out to estimate the total reactive crosssections, product branching ratios, and product quantum state distributions for the O(1D) + HCl reactionusing both reactant and product Jacobi coordinates. The potential energy surface of T. Martinez et al. (Phys.Chem. Chem. Phys.2000, 2, 589) has been used in the calculations. The theoretical predictions are comparedwith experimental results and with the results of classical trajectory calculations on the same surface. Thecomparisons demonstrate the suitability of the potential energy surface and provide useful insights into thereaction mechanism. The calculations using product Jacobi coordinates are the first calculations for this systemwhich permit the prediction of state-to-state reaction probabilities and of product quantum state distributions.

I. Introduction

The investigation of the O(1D) + HCl reaction is motivatedby the importance of this process for the modeling of thechemistry of the atmosphere. Halogenated species react withO(1D), which is produced by the photodissociation of O3. OHand ClO radicals produced in this way can attack O3 and leadto a chain of reactions removing ozone. For this reason, theO(1D) + HCl system has been the subject of several studies.

Experiments include the measurement of the consumptionrate of O(1D),1 the crossed beam production and detection ofClO2 from the O(1D) + HCl f ClO + H process (R1), and thelaser-induced fluorescence (LIF) (including Doppler effects)3-5

and infrared chemiluminescence6 measurements of the vibra-tional distribution of the OH product resulting from the O(1D)+ HCl f OH + Cl reaction (R2).

Theoretical work includes ab initio calculations of points onthe potential energy surface (PES)7-11 as well as quasiclassicaltrajectory (QCT) runs on model12,13 potential energy surfacesand on a PES fitted to ab initio points11,14,15by making use ofpolynomials expressed in terms of bond order (BO) coordi-nates.16 These calculations were unable to reproduce all theproperties derived from experimental measurements. The mostsuccessful of them were those performed on the BO PES (H1)

fitted to ab initio data.11 These were, in fact, able to reproduceall available experimental information except for the productvibrational quantum state distribution of the OH-producingchannel. Recently, some new global PESs (PSB) have beendeveloped for this system by Bowman and collaborators.17-19

These surfaces have been used with success in quantummechanical (QM) wave packet and QCT calculations.19-21 Wavepacket QM calculations have also been performed by Zhangand collaborators23,24 on an old PES12 whose asymptotes havethe wrong energies relative to each other.

Another BO PES (H2)25 has also been produced by fittingan extended set of ab initio data. This new surface is alsoconstrained to have a more appropriate behavior in the areaconnecting the strong interaction region with the asymptotesand is used in the present calculations. The H2 PES featurestwo fairly deep wells (i.e., deep with respect to asymptoticenergy of the entrance channel). The deepest well is associatedwith the insertion of O(1D) into HCl. The other well is associatedwith an attachment of the O atom onto the Cl end of HCl. Thesewells are less pronounced than those of the PSB surface. Thepresence of the wells and the large exothermicity of the reactionmake the energy range spanned by the system quite large.Accordingly, the number of functions which must be includedwhen expanding the system wave function is extremely large.For this reason, to carry out QM calculations, it is moreconvenient to use wave packet techniques based on a gridrepresentation of the wave function. In particular, we chooseto use the real wave packet (RWQM) method.26,27 The results

† Permanent address: Dipartimento di Chimica, Universita` di Perugia,06123 Perugia, Italy.

‡ Permanent address: Departamento de Fısica de la Atmo´sfera, Univer-sidad de Salamanca, 37008 Salamanca, Spain.

5743J. Phys. Chem. A2001,105,5743-5750

10.1021/jp004237t CCC: $20.00 © 2001 American Chemical SocietyPublished on Web 05/30/2001

obtained are then compared with available experimental infor-mation and the result of QCT calculations.

The paper is organized as follows. In section II, we describethe salient features of the wave packet method, while sometechnical details and a discussion of the convergence of thecalculations is given in the Appendix. Sections III-V presentvarious aspects of our results together with comparisons withother theoretical and experimental results. Section VI presentssome conclusions of the work.

II. The Real Wave Packet Approach

Quantum wave packet methods differ from the more tradi-tional time-independent quantum approach in that they are initialvalue methods. That is, the calculation is started from a knownquantum state of the reactants, and the solution of the time-dependent Schro¨dinger equation yields all possible outcomesof interest arising from this starting point. This gives the wavepacket methods the great advantage of calculating state-specificreaction probabilities over the energy range of interest from asingle propagation of the wave packet,26-31 thus facilitating thecalculation of several experimental properties.

In the related numerical procedure, the formalism needed forsuch calculations is attractively simple, and it is possible topropagate only the real part of the wave packet.26,27Furthermore,a particular simple damped Chebyshev iteration, as in the workof Mandelshtam and Taylor,32,33results if one replaces the usualtime-dependent Schro¨dinger equation with a modified forminvolving the arc-cosine of the Hamiltonian operator.26 (Thesame observable properties that would be obtained using theoriginal Schro¨dinger equation are, of course, still obtained withthis modified approach.) For the generic atom-diatom reactions

the state-to-state cross sections and reaction probabilities dependon the energy as well as on the initial vibrotational (V,j) stateof the reactant diatomic molecule BC and on the final vibro-tational (V′,j′) state of the product diatomic molecule AB forprocess 1 (or AC for process 2).

To set up the initial wave packet, we require the wavefunction of the overall system to be expressed in terms of theinitial diatomic molecule BC wave function,æVj

BC(r) (Jacobicoordinates of the reactant arrangementR, r, andΘ are used inthis case). To calculate the total reactive probability and crosssection, one can use reactant coordinates and analyze the wavepacket along a cut corresponding to a large fixed B-Cvibrational coordinate.26,27When the details of product propertiesare needed, it is more appropriate to use product coordinates.Accordingly, if reaction 1 is to be investigated, the wave packetmust be analyzed in terms of the final diatomic molecule ABwave functions,æV′j′

AB(r′), and the primed Jacobi coordinates(R′, r′, and Θ′) of the related arrangement channel are used.Similarly, to analyze reaction 2, the analysis is performed interms of the final diatomic molecule AC wave functions,æV′j′

AC(r′′), and the double-primed Jacobi coordinates (R′′, r′′,andΘ′′) of the related arrangement are used (in the remainderof this section, the formalism will refer only to process 1).

To start the calculations, the initial wave packet in thescattering coordinateR is built up by multiplying together anormalized Gaussian function26 Ne-R(R-R0)2, a phase factor ofthe form e-ik(R-R0), which gives it a relative momentum towardthe interaction region,34 and the vibrational-rotational wave

function of the diatomic reactant. The initial wave packet maytherefore be written as

whereΛ is the quantum number for the projection of the totalangular momentumJ on to the body-fixedz-axis,Pj

Λ(Θ) is thenormalized associated Legendre polynomial, andk is thewavevector which determines the average relative momentumor kinetic energy [Etr° ) (kp)2/2µ] of the collision partners. Notethat in the real wave packet method only the real part of a wavepacket generated by eq 3 is explicitly propagated.26 In thefollowing, since we mostly refer toJ ) 0 calculations, bothJandΛ labels will be dropped when unnecessary.

A grid representation is used to describe the wave packet.The potential and the wave function are represented by theirvalues on a regular grid in the scattering coordinate,R, and inthe vibrational coordinate,r, and on a grid of Gauss-Legendrequadrature points in the Jacobi angle,Θ. At the initial time,the wave packet is placed in the reactant channel, and as timeprogresses, it moves into the interaction region. The initial wavepacket is set up in reactant coordinates. If product quantum statedistributions are sought, the initial wave packet is transformedinto product Jacobi coordinates and the entire propagation iscarried out in these coordinates. The grid must be large enoughto contain the initial wave packet, the region where the analysisline is drawn, and the interaction region. It must also be fineenough to accurately describe the structure of the wave function.At the grid edges, an absorption region is introduced to preventthe wave packet amplitude from reaching the edge of the gridand causing the problem known as aliasing in Fourier transformtheory.30,31

The real part of the wave packet is propagated in time untilit has mainly been absorbed near the edge of the grid. It isanalyzed at every time step along an analysis line in theasymptotic region of the product channel26,27so as to accumulatethe data needed for the computation of the detailedS matrixelementSVjΛ,V′j′Λ′

J (Etr) at the various values of the collisionenergyEtr contained within the wave packet.

By summing the square modulus of the detailedS matrixelements overΛ andΛ′, one can evaluate state-to-state reactionprobabilities PVj,V′j′

J (Etr). A further summation overV′ and j′leads to initial state state-selected reaction probabilitiesPVj

J (Etr).Two types of wave packet calculations are reported in this

paper. The first type is performed using product Jacobicoordinates. These calculations are carried out separately forthe two possible reactions, R1 and R2, and have been performedfor zero total angular momentum only. The advantage of thesecalculations is that they permit the prediction of product quantumstate distributions and branching ratios. The other type ofcalculations is carried out using reactant Jacobi coordinates. Inthese calculations, the objective is to compute just the totalreaction probability as a function of energy. This is done usinga flux analysis method27 in which the outgoing particle fluxthrough a surface at fixed, moderately large values of the HClcoordinate is computed to give the desired result. This methodhas been skillfully adapted in the work of Christoffel et al.20 topermit the prediction of branching ratios, but this approach hasnot been used in the paper presented here. The calculations usingreactant Jacobi coordinates are performed for many values ofthe total angular momentum,J, using a helicity-decoupledapproach, and the results are used to compute approximate totalreactive cross sections.

A + BC(V,j) f AB(V′,j′) + C (1)

A + BC(V,j) f AC(V′,j′) + B (2)

ΨJΛ(R,r,Θ;t)0) ) Ne-R(R-R0)2e-ik(R-R0)æVj

BC(r)PjΛ(Θ) (3)

5744 J. Phys. Chem. A, Vol. 105, No. 24, 2001 Piermarini et al.

III. Reaction Probabilities

Quantum real wave packet calculations were performed forthe reactions

and

on the H2 surface. Reactants were started in their lowestvibrational and rotational states (V ) 0, j ) 0). Each wave packetcalculation yields results for a range of energies. In thecalculations presented here, this range was typically∼0.4 eV.The range is centered around the translational energy,Etr° )(kp)2/2µ (see eq 3), chosen for the wave packet in the reactantchannel. For the calculations reported here,Etr° was chosen tobe between 0.02 and 0.35 eV. This provided reaction prob-abilities over a translational energy range up to 0.52. Details ofthe wave packet calculations, such as grid sizes, absorptionparameters, etc., and discussion of convergence tests are givenin the Appendix. In some cases, more than one calculation wasperformed using different values ofEtr° so as to fully cover theenergy range of interest.

We consider first the total reactive probabilityPV)0,j)0(Etr)summed over all product states and product channels (R1 andR2). This quantity is shown as a solid line in the upper panelof Figure 1. The curves shown in the figure have been computedusing product Jacobi coordinates and zero total angular mo-mentum. All the reaction probabilities, including the totalreaction probabilityPV)0,j)0(Etr), show a sharp rise at threshold,atE ∼ 0.186 eV, which corresponds to zero translational energy.In common with many previous time-dependent reactive scat-tering calculations, the results close to this threshold region areconsidered unreliable and display a sharp peak, which has beenomitted from the figure.34 As the energy increases, the averagevalue of the initial state-selected total reaction probabilityremains substantially constant, while its actual value shows adense structure of peaks. Calculations using reactant Jacobicoordinates lead to very similar curves.

In the central panel of Figure 1, we plot as a solid line thestate-selectedPV)0,j)0 total reactive probability for reaction R1(producing ClO products) and, in the bottom panel, that forreaction R2 (producing OH products). As is apparent from thefigure, the probability for R2 calculated on the H2 potential(solid line of the lower panel) shows a sudden rise at thresholdsimilar to that of the total reactive probability of the upper panel.In this case, however, the maximum is followed by a substantialdecrease at larger energies. For even larger energies, it stabilizesto a value of∼0.4. The corresponding probability for R1 (solidline of the central panel) rises sharply at threshold to a value of∼0.2 and then increases more smoothly to a maximum of∼0.6(around 0.42 eV), after which it gradually decreases. Thissuggests a different mechanism for the two reaction channels.

QCT studies have already highlighted important features ofthe reaction dynamics of the O(1D) + HCl reaction.25 Of keyimportance is the fact that all trajectories explore at least oneof the wells, though for less than a complete rotation. Therefore,even if the action of the wells is unable to enforce a statisticalbehavior, it is a useful paradigm for classifying the reactiveapproach either as attachment (when O attacks on the Cl side)or as insertion. As a matter of fact, at the energy of the crossedbeam experiment,2 one-third of the trajectories react via attach-ment (shorter-lived) and two-thirds by insertion (longer-lived).If the impact parameterb, instead of time, is monitored, otherimportant features of the reaction mechanism are singled out.For both R1 and R2 reactions, attachment processes lead to theusual “hard-sphere like” shape (an opacity function which isnearly constant for smallb values and rapidly decreases to zeroat higherb values). In contrast, while insertion processes leadingto ClO still show the same type of opacity function as attachmentprocesses, the ones associated with the formation of OH showan opacity function exhibiting the less common feature of havinga maximum at a large impact parameter.15,25 This is a sign ofthe high reactivity of H on the H2 PES when attack does notoccur on the Cl end, and this is in accord with the substantiallyheavy heavy light (HHL) nature of R1 as opposed to thesubstantially heavy light heavy (HLH) nature of R2.

The reaction probability for reaction R2 using the H2 PES,given as a solid line in the lower panel of Figure 1, may becompared with that calculated using the PSB PES,20 which areplotted as dots and with those reported by Bitterova et al. in ref19. We see that our calculations predict a sharp rise in thereaction probability at threshold followed by a decrease for theH2 PES. This is characteristic of a barrierless reaction and agreeswell with the results reported in Figure 2 of ref 19. Our analysisof Figure 2 of ref 20, in which an older version of the PSBpotential was used, shows a different form for the thresholdbehavior of the OH+ Cl (R2) channel. Note that there is infact a small barrier to reaction on the H2 surface. There is alsoa strong likelihood that tunneling makes a substantial contribu-tion to the computed reaction probability near threshold. Forreaction channel R1, the probability increases more graduallyabove threshold, implying a less effective contribution oftunneling to this reactive process. This behavior is also observedon the PSB surface.19,20 The possibility of large contributionsof tunneling to reaction for processes involving the exchangeof H with respect to the exchange of heavier atoms (Cl in ourcase) is nicely exemplified in the study of the F+ HD and F+ DH reactive processes of ref 35. Attention should also bedrawn to the comparison of quantum and classical trajectorycalculations presented in Figure 2 of ref 20. This figureillustrates the general good agreement between classical trajec-tory and quantum results and indicates that the contribution of

Figure 1. Total reaction probabilities out of theV ) 0, j ) 0 initialstate plotted vs total energyE. The calculations were performed usingproduct Jacobi coordinates and the real wave packet quantum mechan-ical method with zero total angular momentum: (upper panel) summedover R1 and R2 channels, (middle panel) R1 channel, and (lower panel)R2 channel. For comparison, values obtained by Christoffel et al.20 areshown as dotted lines. These values were estimated from Figure 2 ofref 20.

O(1D) + HCl(V,j) f ClO(V′,j′) + H (R1)

O(1D) + HCl(V,j) f OH(V′,j′) + Cl (R2)

O(1D) + HCl Reaction J. Phys. Chem. A, Vol. 105, No. 24, 20015745

tunneling to the reaction probability in the threshold regionshould be rigorously examined.

The reactive probability versus energy plots of Figure 1exhibit a dense structure of peaks that is similar to both theresults of Bowman and collaborators calculated on the originalPSB surface (results plotted here as dots are interpolated fromthose shown in Figure 2 of ref 20) and the results calculatedusing their new improved surface (see ref 19). Our results donot differ substantially from those given by Bowman andcollaborators:19,20 the global reactive probability rises sharplyat threshold and then stabilizes at higher energies. As alreadymentioned, the largest overall difference between the resultsfor the H2 and PSB surfaces occurs for reaction R1. In thiscase, reaction probabilities calculated using the PSB surface arealways lower than those calculated using the H2 surface, withthe deviation varying from 0.1 to 0.2. The overall agreementof PSB and H2 results for R2 is definitely better.

IV. Branching Ratio and Cross Sections

The ratio of the two probabilitiesPV)0,j)0R1 /PV)0,j)0

R2 for zerototal angular momentum can provide an approximate estimateof the branching ratio between the R1 and R2 reactions. Thisvalue varies from 0.3 in the immediate vicinity of the thresholdto about 1.25 at the higher end of the energy interval that isconsidered. For probabilities calculated on the PSB surface, thisratio varies approximately in the same range (values of ref 20)or is approximately constant around 0.3 (values of ref 19). Allthese values agree with the experimental information of ref 2that gives a lower limit of 0.34( 0.10 for the branching ratioat Etr ) 0.33 eV.

A proper theoretical estimate of the branching ratio requiresthe calculation of the cross sections for both R1 and R2processes. If performed exactly, this calculation would requirethe integration of the quantum mechanical scattering equationsfor all the contributing initial vibrational, rotational, and totalangular momentum quantum numbers. This is still a difficultcomputational task. For this reason, some simplifications havebeen introduced.

QCT calculations indicate that for low reactant rotationalstates the reaction probability is similar to that forj ) 0 (asimplied also by the QM wave packet calculations of ref 19).We consider here only contributions from the ground rotationalstate.

To estimate reliably the total (R1+ R2) reactive cross sectionfrom the initial reactant state (V ) 0, j ) 0), calculations forthe total reaction probability were performed for a large numberof non-zero values of the total angular momentum quantumnumber,J. These calculations were performed in reactant Jacobicoordinates forJ values of 0, 1, 2, 5, 10, 20, 50, 75, 100, 130,150, and 160 using a helicity-decoupling approximation (i.e.,Λ is assumed to equal zero throughout).36 The reactionprobabilities calculated in this way are shown in Figure 2 forJvalues of 0, 50, 100, and 130. The figure is similar in allqualitative respects to Figure 4 of Lin et al.23 Two key featureswhich should be noted from the figure are that, as expected,the threshold for reaction increases with increasing total angularmomentum and that the reaction probability, in the high-energylimit, decreases with increasingJ. The J ) 0 line in Figure 2may be compared with the top panel of Figure 1. The graphsare similar but differ in the positioning of the detailed oscilla-tions. Very many calculations of increasing size have beenperformed to obtain fully converged results for calculations usingboth reactant and product coordinates. The convergence issuesare discussed in the Appendix. While there clearly remain some

aspects of the calculations which are not absolutely converged,all physical quantities reported are stable and are not signifi-cantly changed by improving the calculations, i.e., by increasingthe number of grid points, etc. The integral cross section maybe written as

From the equations, we see that we need the reaction probabilityfor all values ofJ for which it is non-zero at the energy ofinterest. For some values ofJ, we have this quantity from ourquantum calculations. ApproximateJ-shifting37,39,40and relatedcapture model methods36 have been developed to estimate therequired reaction probabilities for those values ofJ for whichactual calculations have not been performed, from probabilitiesfor J values for which they are available. The capture modelmethods were developed to treat situations for which there isno barrier to reaction. In the present case, the threshold toreaction occurs at zero translational energy, indicating that thereis no effective barrier to reaction, and we therefore use amodified capture model method to estimate the total reactivecross section.

In the capture model, for a value ofJ for which we need toestimate the reaction probability, we replacePV)0,j)0

J (Etr) in eq4 with PJh(Etr-Etr

JJh), where PJh(Etr-EtrJJh) is the reaction prob-

ability from the nearest, lower-lyingJh value for which we haveperformed quantum calculations. The probability curve for thisJh value is shifted down in energy by an amountEtr

JJh which isthe difference in the height of the centrifugal barrier forJ andJh (theJ value for which we have actually performed the quantumcalculations). The centrifugal barrier is estimated using aneffective potential which is derived by taking the expectationvalue of the potential over the vibrational-rotational state ofthe reactant diatomic molecule in its initial vibrotational state.In Figure 2, the arrows on thex-axis indicate the energies ofthe barriers on theJ ) 50, 100, and 130 effective potentials. Itis very noticeable that for all the larger angular momenta thereis a substantial reaction probability at energies lower than thesebarriers. This indicates that a reorientation effect, or possiblytunneling, is present in the reaction dynamics.

The probabilities shown in Figure 2 clearly demonstrate thatstandardJ-shifting37,39,40or capture36 models will not work for

Figure 2. Total reaction probabilities out of theV ) 0, j ) 0 initialstate plotted vs total energyE for four different total angular momenta,J. The calculations were performed using reactant Jacobi coordinatesand the real wave packet quantum mechanical method.

σ(Etr) )π

kVj2∑J)0

∝

(2J + 1)PV)0,j)0J (Etr) (4)

5746 J. Phys. Chem. A, Vol. 105, No. 24, 2001 Piermarini et al.

this case. This is because not only is the reaction probabilityshifted to higher energies with increasing total angular momen-tum, but its magnitude is also decreased. Bittererova et al.19

have elegantly analyzed this problem and have developed asophisticated, but difficult to use, newJ-shifting procedure.38

Our approach to this problem has been first to compute thereaction probabilities for a large set of total angular momentumvalues (J values) spanning nearly the entire range whichcontributes to the reaction cross section. This is up to 175 forthe energies we consider. If we then need the reaction probabilityfor a J value between twoJ values (J1 and J2) for which wehave already computed the exact reaction probabilities, weproceed as follows.

(1) Use the capture model to estimatePJ(Etr) starting fromthe lower of the two total angular momentaJ1. This gives usPJ(Etr-Etr

J1J).(2) Use the capture model to estimate the same quantity, but

now starting from the reaction probability calculated usingJ2.This gives usPJ(Etr-Etr

J2J) (note thatEtrJ2J is negative).

(3) Now estimate the reaction probability for total angularmomentum J, PJ(Etr), by interpolating between these twoestimates:

This procedure is in fact the same as that used by Gray et al.41

The estimated total reaction cross section is shown in Figure3. These results may be compared with the most recent resultsof the Bowman group (Figure 11 of ref 19). The total reactivecross sections predicted using the H2 surface (i.e., the calcula-tions presented here) are clearly larger, by a factor of∼1.8,than those computed using the improved PSB potential. Thisdoes not arise from significantly greater reaction probabilitiesfor a particular value ofJ but must rather arise from a largerrange ofJ values contributing to the cross section (see eq 4).

V. Product Distributions

Additional indications concerning the suitability of theproposed PES are provided by a comparison of calculatedenergetic distributions of the products with experimental data.Information about the product vibrational distribution (PVD)of R2 comes from the infrared chemiluminescence experiment6

already mentioned. To better understand the main features ofthe PVDs, these were calculated for both R1 and R2 processesat different translational energy values. Some of these distribu-

tions are shown in Figure 4, after normalization to a maximumvalue of 1.0.

The central panel of Figure 4 compares the PVD calculatedatEtr ) 0.26 eV using the real wave packet quantum mechanical(RWQM) method with the experimentally determined quantitymeasured for collision energies ranging from 0 to 0.26 eV.6

Both distributions are normalized to unity at their maximumvalue. The calculated and experimental PVDs agree fairly well,both being inverted with the RWQM distribution peaking atV′) 2, one vibrational quantum less than the measured one. QCTresults have not been published at 0.26 eV. However, as shownin the right-hand panel of the figure, where the various PVDscalculated on both H2 and PSB PESs for R2 at 0.53 eV areplotted, the two quasiclassical results are substantially identicalsince they both peak atV′ ) 2 (with V′ ) 3 being the nextmore populate state) and die off atV′ ) 6. In contrast, theRWQM PVD is clearly much hotter since it has an absolutemaximum atV′ ) 4 and a secondary one atV′ ) 2. The inversionof the PVD for OH is made stronger by an increase in thecollision energy. The same is true for the transfer of Cl. Forthis process, as shown in the left-hand panel of the same figure,the maximum of the calculated PVD gradually shifts fromV′) 0 to V′ ) 3 as the energy increases from 0.26 to 0.53 eV.

A further comparison with the experiment can be carried outin terms of the translational energy distribution of the products.The product translational distribution (PTD) for the ClO-formingprocess was derived from the measurements of ref 2. In theleft-hand panel of Figure 5, we compare the PTD for R1calculated using the RWQM method with aJ of 0 atEtr ) 0.53eV on the H2 surface with the QCT ones obtained on both H2(- - -) and PSB (-‚‚-), and experimental information (‚‚‚).To make the comparison more homogeneous, we have boxedRWQM results that are by nature discrete (boxes of size 0.22and 0.05 eV were used for OH and ClO, respectively). Again,the agreement between QCT results obtained on the two surfacesis fairly good, and the agreement with experimental data is alsoexcellent. The agreement with the RWQM PTD is on theaverage fairly good. A peculiarity of the quantum PTD is thestructure that reflects the product vibrational state distribution

Figure 3. Total reactive cross section calculated using a modifiedcapture model plotted as a function of collision energyEtr.

PJ(Etr) ) PJ(Etr-EtrJ1J)( J2 - J

J2 - J1) + PJ(Etr-Etr

J2J)( J - J1

J2 - J1)(5)

Figure 4. Normalized product vibrational distributions. In the middlepanel are shown RWQM H2 (s) OH product vibrational statedistributions calculated at a collision energy of 0.26 eV for reactantsinitially in their lowest vibrational-rotational state. For comparison,experimental data6 (‚‚‚) are also shown. In the right-hand panel areshown RWQM H2 (s), QCT H2 (- - -), and QCT PSB (‚--‚)20

OH product vibrational state distributions calculated at a collision energyof 0.53 eV. In the left-hand panel are shown QCT H2 OCl productvibrational state distributions calculated at a collision energy of 0.26(- - -), 0.33 (s), and 0.53 eV (‚‚‚).

O(1D) + HCl Reaction J. Phys. Chem. A, Vol. 105, No. 24, 20015747

with which it is associated. A finer resolution of measured datacould, possibly, improve the quality of the comparison.

In the right-hand panel of the Figure 5, PTDs which werecalculated using QCT and RWQM methods on the H2 potentialenergy surface for the OH+ Cl channel (R2) are presented.Apart for the fine structure of the quantum distribution, theagreement between the QCT and RWQM results is alsosatisfactory in this case. It should be remembered that thequantum mechanical calculations are performed for zero totalangular momentum and for reactants in their lowest vibrational-rotational state. In contrast, the QCT calculations include alltotal angular momenta and sample a thermal distribution ofinitial quantum states. This means that it is not really possibleto make detailed comparisons between these two types ofcalculations.

VI. Conclusions

Time-dependent quantum calculations performed on the H2PES suggest that the quantum contribution to reactivity atthreshold may be important. As a matter of fact, the energydependence of the total reaction probability for both reactiveprocesses has a negative slope at (or maybe just marginallyabove) threshold. Reaction R2 has a higher probability atthreshold, and this may arise from a tunneling contribution atthese low energies.

The reaction probabilities which result directly from thequantum mechanical calculations are not experimentally mea-surable. To make predictions about the total reactive crosssection, which is an observable quantity, quantum mechanicalreactive scattering calculations were performed using reactantJacobi coordinates within the helicity decoupling approximationfor a wide range of total angular momenta (J ) 0, 1, 2, 5, 10,20, 50, 75, 100, 130, 150, and 160). These calculations haveenabled us to reliably predict the total reactive cross section asa function of collision energy.

In this paper, we have presented the first state-to-statequantum reactive scattering calculations for the O(1D) + HClsystem. These were performed using product Jacobi coordinatesand have permitted us to make predictions about productquantum state distributions for the system. Product vibrationalstate and translational energy distributions, whose detailed natureis a critical test of the accuracy of the PES, show that quantum

effects are important for these reactions, showing significantdifferences between QCT and QM results. In particular, wefound that QCT results obtained on two different PESs lead tosubstantially identical vibrational distributions. On both surfaceswhen moving to higher energies, these distributions becomebroader and shift their maximum to higher vibrational quantumnumbers. However, QM distributions always have a peakdisplaced to a higher quantum number.

Acknowledgment. Thanks are due to the EU COST programin chemistry, action D9, for supporting a stay of V.P. at Bristolwithin the short term mobility program. Financial support fromMURST, ASI, and CNR is also acknowledged. We thank theEPSRC for the provision of funds to purchase the computerson which this work was performed and for funds to underpinthe collaboration under the COST program. S.K.G. was sup-ported by the Office of Basic Energy Sciences, Division ofChemical Sciences, U.S. Department of Energy, under ContractW-31-109-ENG-38. F.G. thanks TUBITAK for financial supportvia Grant 1739. M.L.H. thanks DGESIC of Spain for financialsupport (Grant PB98-0281).

Appendix

Details of Quantum Wave Packet Calculations andDiscussion of Convergence.In this Appendix, details of thewave packet calculations are presented and a discussion of theconvergence of the results is given. Three types of wave packetcalculations were performed to obtain the results presented inthe body of the paper. These were (1) calculations using OH-Cl product Jacobi coordinates; (2) calculations using OCl-Hproduct Jacobi coordinates, and (3) calculations using O-HClreactant Jacobi coordinates.

The details of the calculations using OH-Cl and OCl-Hproduct Jacobi coordinates are presented in Table 1. Thedamping operator,26 A, used for the product coordinate calcula-tions corresponds to an exponential imaginary absorbingpotential.34,42,43 The detailed form of the operator isAx(x) )exp{-cabsexp[-2(xmax - xabs)/(x - xabs)]} for x > xabsandAx

) 1, and otherwise forx ) R′ or r′. The calculations wereentirely “grid-based”. This means that the wave packet wasrepresented on an angular grid based on Gauss-Legendrequadrature points44,45and the effective number of angular basisfunctions is equal to the number of angular quadrature points.The state-to-state reaction probabilities are obtained by analyzingthe wave packet at each time step26 along a cut at a large fixedvalue of the product scattering coordinate,R∞′.

Figure 5. Product translational distributions calculated atV ) j ) 0and a collision energy of 12.2 kcal/mol. In the left-hand panel are shownRWQM H2 (s), QCT H2 (- - -), QCT PSB (-‚‚-), and measured(‚‚‚) distributions for R1. In the right-hand panel are shown RWQMH2 (s) and QCT H2 (- - -) distributions.

TABLE 1: Grid and Initial Condition Details forWavepacket Calculations Using OH-Cl and OCl-HProduct Jacobi Coordinatesa

OH-Cl OCl-H

scattering coordinate (R′) range/a0 0-15 0-15no. of grid points inR′ 251 251internal coordinate (r′) range/a0 0-12 0-14no. of grid points inr′ 251 229no. of angular grid points 94 100absorption region length inR′ (r′)/a0 4(2) 4(4)absorption strength (cabs)b 0.1 0.1center of initial (sinc) wave packet34 (R0)/a0 7 7width of the wave packet,R 8.0 8.0smoothing of the wave packet,â 0.2 0.2initial translational energy,Etr (eV) 0.08, 0.35 0.08, 0.35position of analysis line,R∞′ 10 10

a All quantities are given in atomic units.b The damping operatorAused corresponded to an exponential imaginary absorbing potential.See appendix for details.

5748 J. Phys. Chem. A, Vol. 105, No. 24, 2001 Piermarini et al.

The details of the calculations using O-HCl reactant Jacobicoordinates are presented in Table 2. The damping operator,26

A, used for these calculations corresponds to a quadraticimaginary absorption potential.42,43 For the calculations usingreactant coordinates, our interest is in the total reactionprobability for each value ofJ that is used. For these calcula-tions, we have used a flux analysis method,27 and the analysisline was located at a large fixed value of the H-Cl vibrationalcoordinate. For the reactant coordinate calculations, the com-puter code used angular basis functions, rather than grid points,as the primary representation.

Each wave packet calculation in product coordinates typicallytakes more than 1 week to complete on our SGI/R10000 Origin200 computer, while those using reactant coordinates take∼2days. Very many test calculations have been carried out,spanning a period of more than 1 year, to test and confirm theresults reported in the paper.

Figure 6 compares the total reaction probability for aJ of 0calculated using reactant and product coordinates. The totalreaction probability calculated using product reaction coordinatesis in fact the sum of two independently computed reactionprobabilities, one for the OH+ Cl product and one for the OCl+ H product. The agreement between reaction probabilitiescalculated using reactant and product coordinates is not nearlyas good as that which has been obtained for the O+ H2

system.34 This reflects the greater difficulty of performingquantum mechanical calculations on systems containing heavynuclei. The cross section for the reaction is the result of summingover reaction probabilities for manyJ values (see eq 4). The

result of this can be seen in Figure 3, where the result ofsumming over manyJ values has completely smoothed out theoriginal oscillations in the reaction probabilities. The mostimportant aspect of the reaction probabilities as far as thereactive cross sections are concerned is therefore the averagevalue of the reaction probability over a range of energies, ratherthan the detailed oscillation which the reaction probabilitydisplays on a fine energy scale. The average of the productcoordinate total reaction probability over the energy range of0.23-0.6 eV is 0.845, while the same average for the reactantcoordinate calculations is 0.804. The difference in the averagereaction probabilities over the range of the calculations is 5%,and this represents a fair assessment of the uncertainty in theresults presented here.

We note that these are the very first calculations presentedfor the system in which product Jacobi coordinates have beenused, and therefore also the first calculations which have beenable to address the product quantum state distribution in thereaction. No comparisons of the type we present here havebefore been possible for this system. Our reactant coordinatecalculations are similar in specification to those used in someother publications.20 Lin et al.23 however have used a far largernumber of angular basis functions. We have performed calcula-tions with up to 120 angular basis functions to check the resultspresented here. Such calculations result in changes to the detailedfine structure in the reaction probability versus energy graphs,but they do not significantly change the average value of thereaction probability, which is the essential quantity whichdetermines the observable quantity, namely, the total reactioncross section.

The question of the reliability of the product quantum statedistributions predicted by our calculations then arises. In anattempt to examine this question, we have computed OH productvibrational quantum state distributions from two independentwave packet calculations which used different numbers ofangular basis functions, 80 and 94. The results are shown inFigure 7. The maximum difference between the two sets ofproduct state distributions is 5%, confirming our previousestimate of the reliability of our results.

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TABLE 2: Grid and Initial Condition Details forWavepacket Calculations Using O-HCl Reactant JacobiCoordinatesa

scattering coordinate (R) range/a0 0-14no. of grid points inR 251internal coordinate (r) range/a0 0-11.5no. of grid points inr 251no. of angular grid points 127absorption region length inR (r)/a0 4(4)absorption strength (cabs)b 0.005center of initial (sinc) wave packet34 (R0)/a0 9.5width of the wave packet,R 0.4initial translational energy,Etr (eV) 0.08, 0.35position of analysis line,r∞ 6

a All quantities are given in atomic units.b The damping operator,A, used corresponded to a quadratic imaginary absorbing potential. Seethe Appendix for details.

Figure 6. Comparison of total reaction probability computed usingreactant (s) and product (‚‚‚) Jacobi coordinates.

Figure 7. Comparison of OH product vibrational distributions usingresults of two different product Jacobi coordinate calculations. The twocalculations used different numbers of angular basis functions: (s)94 angular basis functions and (- - -) 80 angular basis functions.

O(1D) + HCl Reaction J. Phys. Chem. A, Vol. 105, No. 24, 20015749

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